LHS = {1 + cotx – (-cosecx)} {1 + cotx + (-cosecx)} = {1 + cotx + cosecx} {1 + cotx – cosecx} = {(1 + cotx) + (cosecx)} {(1 + cotx) – (cosecx)} We know that (a + b) (a – b) = a 2 – b 2 = (1 + cotx) 2 – (cosecx) 2 = 1 + cot 2 x + 2 cotx – cosec 2 x We know that 1 + cot 2 x = cosec 2 x = cosec 2 x + 2 cotx – cosec 2 x = 2 cotx = RHS Hence proved.

Prove that :

LHS

= {1 + cotx – (-cosecx)} {1 + cotx + (-cosecx)}

= {1 + cotx + cosecx} {1 + cotx – cosecx}

= {(1 + cotx) + (cosecx)} {(1 + cotx) – (cosecx)}

We know that (a + b) (a – b) = a² – b²

= (1 + cotx)² – (cosecx)²

= 1 + cot²x + 2 cotx – cosec²x

We know that 1 + cot²x = cosec²x

= cosec²x + 2 cotx – cosec²x

= 2 cotx

= RHS

Hence proved.